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# Casino Math Explained

When it comes to casino math, one arm of mathematics is more important than the rest: probability theory. Probabilities are always assessed when creating a new gambling game or adapting it to a new variant. This way, casinos can ensure that with any bet, there’s always a house edge, and the game can’t be defeated through pure math.

Probability is primarily assessed in casinos as expected value. By assessing the probabilities of all of the bets in a game, the house can predict future losses or gains by multiplying all the probabilities of each potential outcome by the payoff value. This way, a casino can always function as a business as profit is mathematically assured.

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If the casino can use math to its advantage, can players? Casino mathematics concern profit over time, but player math and gambling mathematics can potentially help with quick gains if sound probability mathematics are applied.

## Fundamentals of Probability

Probability underpins all casino math. So to explain casino math, we first need to explain the fundamentals of probability.

Being between partly and wholly randomized, every casino game starts a round or run with the same probabilities for each outcome. This can swing a bit with live card games drawing from the deck, but over a phase of that game in which part of the shoe is used and then changed, the probabilities remain mostly the same. These probabilities are then converted into odds for the players that closely, but not exactly, reflect the probability of each bet winning.

### Basics of Probability Theory

In its simplest form, probability theory is the mathematical framework that lets us systematically perform an analysis of the likelihood of outcomes in a given event.

Probability is usually expressed as a value between 0 and 1, with 1 being a certainty. In casino games, all of the potential outcomes are taken, and then the likelihood of each one is calculated based gambling statistics and on probability theory. This often informs the odds offered.

A prime example of probability applied to popular casino games is with inside betting on roulette. You can bet on 37 numbers (0-36), and if your straight bet wins, you get a payout of 35:1. The probability is 37:1, but the payout returns 36 from one coin. So, the odds are weighed slightly under the probability to allow for a house edge.

In the card game of baccarat, you only have three betting zones: player, banker, and tie. The probability of a tie occurring is around 0.095, with odds of 8:1 paying nine coins from a one-coin bet. For the standard bets, the banker has a likelihood of 0.457 of winning, while the player is at 0.422. So, the banker payout is cut to 0.95:1 to the player’s 1:1.

## House Edge Demystified

You will have seen the term “house edge” mentioned a few times already. It’s a key value in casino math, especially because it shows how well the payouts reflect the true odds of probability.

### Understanding House Edge

House edge is shown as a percentage. This percentage represents the average gross profit that a game is expected to make for a casino over time. The house edge can even be calculated as a figure in set bets and not just for games as a whole. All casino games have this built-in edge that allows the house to run as a business.

### Factors Influencing House Edge

The main factors affecting house edge are the probabilities of any bet coming in and the odds offered when that particular bet wins. There will always be a discrepancy in the exchange from probability to payout. In a roulette wheel, you can bet on red or black for 1:1 payouts, but the probability of winning is less than 50-50 due to the green zero pocket. If those odds were based on probability, the payouts would be slightly higher.

## Random Number Generators (RNGs)

In the vast majority of the most popular games at an online casino, a random number generator dictates the play. Being able to rely on a trusted randomized program has allowed players to enjoy digital casino gaming, but does it influence probability?

### Role of RNGs in Casino Operations

Random Number Generators (RNGs) have famously been a staple of slot machines, from their standing-form one-armed bandits to online slots. The role of RNGs in slots and video form table games is to ensure that every outcome is wholly randomized, as the name suggests. They will be programmed in certain ways for slots to, on average, reflect factors like the return to play (RTP) and volatility, but the results seen will still be randomized for each player.

### Ensuring Fairness and Randomness

To ensure the fairness and randomness of each RNG-powered casino game, developers have to submit their games for auditing and playtesting. Third-party agencies and regulators are relied upon to certify the randomness of the RNG program. The games won’t be permitted under respected regulators to come into circulation if it fails.

Random number generators are, for the most part, pseudo-random number generators as they have to rely on a set of rules in their algorithms to attempt to replicate true randomness. Recently, quantum random number generators have been getting a lot of press.

This is because their random outputs are generated by sampling a signal of large quantum dynamic numbers. In theory, these QRNGs can offer an enhanced level of randomness. Perhaps one day, these will power our online games like Blackjack 23+1.

## Statistics and Casino

Statistics are often sought out by players to inform their bets. Where possible, many will look to recent outcomes in the more or wholly randomized games to find hot or cold games or bets. This is folly as each round is randomized all over again. Still, some statistical models do play into the casino’s setup.

### The Law of Large Numbers

The law of large numbers suggests that, with a large sample of random events, the results will reflect the given probability of the outcomes. On a coin toss performed enough times, the theory states that it will land on heads an equal number of times to tails. For the house, even losing to a big bet is equalled out or outdone by the house edge bringing in a sustained profit over a large sample.

### Central Limit Theorem

Central limit theorem offers the premise that, with a “sufficiently large” sample size from a population with a set amount of variance, the mean found from the sampled variables from that population will be close to equal that of the mean of the whole population.

For casinos, it means that with randomly produced results, with enough plays, the observed averages of each outcome will closely approximate the normal distribution.

## Other Casino Math Concepts

Many different math concepts have rightly, and sometimes wrongly, been applied to casino games. Here are some more of the big ones.

### Expected Value

Expected value can represent the average amount you can expect to win on each bet should you bet at identical odds several times over. It’s calculated by the sum of the probability of each outcome multiplied by its payout (or value).

### Combinations and Permutations

Combinations and permutations are a big part of blackjack and poker (relating to winning combinations of the cards in the hand and on the table), the outcomes of two dice being rolled in craps, and even the symbol combinations that can occur on a slot. Working out these permutations in gambling often comes down to an exercise in combinatorial calculus, in which you work out the probability of set combinations occurring.

### The Binomial Method

The binomial method is a statistical method of common discrete distribution. The method involves performing a trial multiple times in which an outcome can either succeed or fail (win or lose) as a bet, for example. So, binomial distribution offers a probability for success across a set number of trials and the probability for success in each trial.

## Casino Math FAQs

### What math is used in casinos?

Many mathematical formulas go into a high-functioning casino. More fundamentally for casino games, probability theory is core to all math in play in most games. You also have expected value, which works out the likelihood of any bet winning on a given game.

### What is the formula for probability in casino gaming?

The formula for probability in casino gaming is the number of favourable outcomes from your bet divided by the total number of possibilities. So, the formula can be expressed as favourable outcomes (f) divided by total possible outcomes (t) equals probability (p), or f / t = p

### Can you use math to win in casinos?

In theory, you can use math to get an edge on casinos. Blackjack is the prime example of using math or more specifically, probability, to win in a casino. While the practice of counting cards is heavily frowned upon and will get you banned from land-based casinos, applying basic strategy, which is based on probability, is perfectly acceptable. It doesn't guarantee wins, but it does offer a path to optimized play according to the principles of probability. This is the best that you can hope for in a game that's randomized.

### What is the only mathematically beatable game in a casino?

Under the right conditions and with a lot of skill, blackjack is the only mathematically beatable game in a casino. You can use the actions laid out by basic strategy to make the correct plays every time, but even this doesn't guarantee a win with every hand. Some can count cards to have an even better idea of which cards will likely be coming out of the shoe next, but this practice will get you banned from casinos.